arXiv research

The paper studies invariant weighted Bergman metrics on domains.

Survey of nonnegative scalar curvature sequences and their limits.

Tian's theorem applies to Moishezon spaces with singular metrics.

In this paper, we give a lower bound of Bergman kernels for a sequence of almost Kähler-Einstein Fano manifolds, or more general, a sequence of Fano manifolds with almost Kähler-Ricci solitons. This generalizes a result by Donaldson-Sun, Tian for Kähler-Einstein manifolds sequence with positive scalar curvature. As an …

We prove that the Yang-Mills $α$-functional satisfies the Palais-Smale condition. This guarantees the existence of critical points, which are called Yang-Mills $α$-connections. It was shown by Hong, Tian and Yin in [10] (to appear in Comm. Math. Helv.) that as $α\to 1$, a sequence of Yang-Mills $α$-connections converge…

Paper proves Hamilton-Tian conjecture using partial C0-estimate.

Tian's theorem connects Chern classes of bundles to random section zeros and degeneracy sets.

Solves Tian's stabilization problem for toric Fano manifolds.

In this note, using the recent compactness results of Tian and Chen-Donaldson-Sun, we prove the K-semistable version of Yau-Tian-Donaldson correspondence for Fano manifolds.

The paper solves a uniform Yau-Tian-Donaldson conjecture for toric manifolds.

Proves Yau-Tian-Donaldson conjecture for toric manifolds and bundles.

In this paper, we explore the limit structure of a sequence of Riemannian manifolds with Bakry-Émery Ricci curvature bounded below in the Gromov-Hausdorff topology. By extending the techniques established by Cheeger-Cloding for Riemannian manifolds with Ricci curvature bounded below, we prove that each tangent space at…

New examples of manifolds with positive scalar curvature and infinitely many poles.

In this work, we describe the asymptotic behavior of complete metrics with prescribed Ricci curvature on open Kahler manifolds that can be compactified by the addition of a smooth and ample divisor. First, we construct a explicit sequence of Kahler metrics with special approximating properties. Using those metrics as s…

We give a new Tian-Todorov lemma on deformations of CR-structures and use it to reprove the deformation unobstructedness of normal compact strongly pseudoconvex CR-manifold under the assumption of $d'd''$-lemma, more faithfully following Tian-Todorov's approach.

This an announcement for the generalized asymptotic expansion of Tian-Yau-Zeldtich.

Paper confirms Hamilton-Tian conjecture for specific Sasakian manifolds.

In this paper, we compute the Tian-Zhu invariant on hypersurfaces of complex projective spaces.

Uniform proof of Kähler-Einstein metrics with arbitrary polarizations.

Solves Yau-Tian-Donaldson conjecture for smooth projective varieties.

In [Cheeger-Tian 2005], Cheeger-Tian proved an $ε$-regularity theorem for $4$-dimensional Einstein manifolds without volume assumption. They conjectured that similar results should hold for critical metrics with constant scalar curvature, shrinking Ricci solitons, Ricci flows in $4$-dimensional manifolds and higher dim…

We introduce a new continuity method which provides an alternative way of carrying out the Analytic Minimal Model Program introduced by G. Tian and J. Song and G. Tian. This equation -- unlike the Ricci flow -- has the advantage of having Ricci curvature bounded from below along the deformation, so that the compactness…

In this paper, we study the behavior of Bergman kernels along the Kähler Ricci flow on Fano manifolds. We show that the Bergman kernels are equivalent along the Kähler Ricci flow under certain condition on the Ricci curvature of the initial metric. Then, using a recent work of Tian and Zhang, we can solve a conjecture …

Effective Yau-Tian-Donaldson conjecture for spherical varieties.

We give a variational proof of a version of the Yau-Tian-Donaldson conjecture for twisted Kähler-Einstein currents, and use this to express the greatest (twisted) Ricci lower bound in terms of a purely algebro-geometric stability threshold. Our approach does not involve the continuity method or Cheeger-Colding-Tian the…

Counterexample disproves conjectures about log canonical thresholds.

Proves algebraic version of Hamilton-Tian conjecture for log Fano pairs.

We study Tian's $α$-invariant in comparison with the $α_1$-invariant for pairs $(S_d,H)$ consisting of a smooth surface $S_d$ of degree $d$ in the projective three-dimensional space and a hyperplane section $H$. A conjecture of Tian asserts that $α(S_d,H)=α_1(S_d,H)$. We show that this is indeed true for $d=4$ (the res…

In this paper, we compute Tian's $α_{m,k}^{K\times K}$-invariant on a polarized $G$-group compactification, where $K$ denotes a maximal compact subgroup of a connected complex reductive group $G$. We prove that Tian's conjecture (see Conjecture 1.1 below) is true for $α_{m,k}^{K\times K}$-invariant on such manifolds wh…

Associated with a smooth, $d$-closed $(1, 1)$-form $α$ of possibly non-rational De Rham cohomology class on a compact complex manifold $X$ is a sequence of asymptotically holomorphic complex line bundles $L_k$ on $X$ equipped with $(0, 1)$-connections $\bar\partial_k$ for which $\bar\partial_k^2\neq 0$. Their study was…

Study Bergman kernels on Kähler manifolds, answering Lu-Tian's question.

Study geometric operators on Tian-Yau spaces, finding $L^2$ harmonic forms and asymptotic regularity.

Proves Yau-Tian-Donaldson conjecture for cohomogeneity one manifolds.

We provide here a counter-example to the second inequality of Corollary (19.10) in the Clay Institute Monograph by J.Morgan and G.Tian entitled "Ricci Flow and the Poincare Conjecture". We had announced the existence of this counter-example in our paper "Five Gaps in Mathematics", Advanced Non-linear Studies, vol 15, N…

In the present paper and the companion paper [8] a probabilistic (statistical mechanical) approach to the study of canonical metrics and measures on a complex algebraic variety X is introduced. On any such variety with positive Kodaira dimension a canonical (birationally invariant) random point processes is defined and…

Establishes Yau-Tian-Donaldson conjecture for weighted metrics.

Proves existence of Kähler-Einstein metrics in big cohomology classes.

Well-known conjectures of Tian predict that existence of canonical Kahler metrics should be equivalent to various notions of properness of Mabuchi's K-energy functional. In some instances this has been verified, especially under restrictive assumptions on the automorphism group. We provide counterexamples to the origin…

Research confirms Streets-Tian conjecture for 2-step solvmanifolds.

In this paper is to extend the Cheeger-Colding Theory to the class of conic Kahler-Einstein metrics. This extension provides a technical tool for [LTW] in which we prove a version of the Yau-Tian-Donaldson conjecture for Fano varieties with certain singularity.

The global holomorphic α-invariant introduced by Tian is closely related with the study in the existence of Kahler-Einstein metric. We apply the result of Tian, Lu and Zelditch on polarized Kahler metrics to approximate plurisubharmonic functions and compute the α-invariant of toric Fano manifolds.

This paper constructs and studies the Gromov-Witten invariants and their properties for noncompact geometrically bounded symplectic manifolds. Two localization formulas for GW-invariants are also proposed and proved. As applications we get solutions of the generalized string equation and dilation equation and their var…

In this paper, we generalize Chen-Tian energy functionals to Kähler-Ricci solitons and prove that the properness of these functionals is equivalent to the existence of Kähler-Ricci solitons. We also discuss the equivalence of the lower boundedness of these functionals and their relation with Tian-Zhu's holomorphic inva…

Study proves Yau-Tian-Donaldson conjecture for generalized Kähler-Ricci solitons.

New proofs of Donaldson-Uhlenbeck-Yau theorem using geodesic rays.

We survey some recent developments in the direction of the Yau-Tian-Donaldson conjecture, which relates the existence of constant scalar curvature Kähler metrics to the algebro-geometric notion of K-stability. The emphasis is put on the use of pluripotential theory and the interpretation of K-stability in terms of non-…

We generalize the maximal time existence of Kähler-Ricci flow in Tian-Zhang and Song-Tian to conical case. Furthermore, if the twisted canonical bundle $K_{M}+(1-β)[D]$ is big or big and nef, we can expect more on the limit behaviors of such conical Kähler-Ricci flow. Moreover, the results still hold for simple normal …

We prove that, on a minimal elliptic Kähler surface of Kodaira dimension one, the continuity method introduced by La Nave and Tian in \cite{LT} starting from any initial Kähler metric converges in Gromov-Hausdorff topology to the metric completion of the generalized Kähler-Einstein metric on its canonical model constru…